#9062227

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26,29,33)(27,30,35,31)(28,32,37,40)(34,38,36,39);(1,2)(3,7)(4,5)(6,8)(10,11)(15,17)(16,18)(22,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(2,5)(3,8)(10,11)(12,19)(13,21)(22,23)(25,27,29,35)(26,31,33,30)(28,36,37,34)(32,38,40,39);(1,3,4,8)(2,6,5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,29,37)(26,32,33,40)(27,34,35,36)(30,38,31,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969423);